There used to be a link to a set of documentation on how to set up Jenkins on Google App Engine and configure it for push-to-deploy functionality, the original link no longer works however, it is still on the wayback machine (click here to view).
Other cloud-based solutions either integrate with Bitbucket, or integrate with GAE but not both, which has led me to evaluating setting up my own Jenkins instance.
Are the instructions provided in the link still recommended? Or have they been taken down because there is a newer option that I can't locate?
Push to Deploy, as you correctly noted, isn't a part of the Cloud Platform any longer, but you can still use Compute Engine or Container Engine (another link) to build a Jenkins pipeline which responds to git push by building and deploying using webhooks or similar functionality.